Rockafellar et al. (2014) developed a new CVaR (Superquantile) quadrangle with the Statistic equal to CVaR. This theoretical result is a basis for the CVaR (Superquantile) regression suggested in Rockafellar et al. (2014). CVaR regression is done by minimizing CVaR (Superquantile) Error, which is an element of CVaR (Superquantile) quadrangle.

Alternatively, CVaR regression can be done in two steps (Rockafellar et al (2008)):

Step 1. Minimization of CVaR Deviation from CVaR (Superquantile) Quadrangle with the residual depending only upon loading factors.

Golodnikov et al. (2016) suggested two sets (Set 1, and Set 2) of parameters for the Mixed-Quantile-Based Quadrangle. Set 1 corresponds to the Linear Programming implementation of the CVaR (Superquantile) regression considered in Rockafellar et al. (2014).

Golodnikov et al. (2016) showed that the minimization of CVaR (Superquantile) Error for a discrete distribution can be reduced to the minimization of Rockafellar Error from the Mixed-Quantile-Based Quadrangle with the Set 1 of parameters. Therefore, CVaR (Superquantile) regression can be done by minimizing the Rockafellar Error of the residual depending upon intercept and loading factors.

Moreover, for a discrete distribution, Golodnikov et al. (2016) showed that the Deviation in the Mixed-Quantile-Based Quadrangle (called Mixed CVaR Deviation) with the Set 1 and Set 2 of parameters coincides with the Deviation in CVaR (Superquantile) quadrangle. Therefore, CVaR (Superquantile) regression can be done with the two-step procedure.

This case study implemented the following equivalent variants of CVaR (Superquantile) regression:

• Minimization of CVaR (Superquantile) error with PSG functioncvar2_err.
• Two step procedure with CVaR Deviation from CVaR (Superquantile) Quadrangle.
• Minimization of Rockafellar Error Function, ro_err, with the Set 1 of parameters.
• Two step procedure using the mixed CVaR Deviationwith Set 1 and Set 2 of parameters.